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arXiv · 1311.4041

The Mean Square of Divisor Function

Chaohua Jia·Ayyadurai Sankaranarayanan
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Abstract

Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $$\sum_{n\leq x}d^2(n)=xP(\log x)+E(x), $$ where $P(y)$ is a cubic polynomial in $y$ and $$ E(x)=O(x^{{3\over 5}+ε}), $$ where $ε$ is a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis(RH), $$ E(x)=O(x^{{1\over 2}+ε}). $$ In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $$ E(x)=O(x^{1\over 2}(\log x)^5\log\log x). $$ In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we shall prove $$ E(x)=O(x^{1\over 2}(\log x)^5). $$

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Chaohua Jia, Ayyadurai Sankaranarayanan. 2014-03-23. The Mean Square of Divisor Function. https://arxiv.org/abs/1311.4041

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